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Theorem dvelimdc 2179
Description: Deduction form of dvelimc 2180. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
dvelimdc.1 xφ
dvelimdc.2 zφ
dvelimdc.3 (φxA)
dvelimdc.4 (φzB)
dvelimdc.5 (φ → (z = yA = B))
Assertion
Ref Expression
dvelimdc (φ → (¬ x x = yxB))

Proof of Theorem dvelimdc
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 nfv 1402 . . 3 w(φ ¬ x x = y)
2 dvelimdc.1 . . . . 5 xφ
3 dvelimdc.2 . . . . 5 zφ
4 dvelimdc.3 . . . . . 6 (φxA)
54nfcrd 2173 . . . . 5 (φ → Ⅎx w A)
6 dvelimdc.4 . . . . . 6 (φzB)
76nfcrd 2173 . . . . 5 (φ → Ⅎz w B)
8 dvelimdc.5 . . . . . 6 (φ → (z = yA = B))
9 eleq2 2083 . . . . . 6 (A = B → (w Aw B))
108, 9syl6 29 . . . . 5 (φ → (z = y → (w Aw B)))
112, 3, 5, 7, 10dvelimdf 1874 . . . 4 (φ → (¬ x x = y → Ⅎx w B))
1211imp 115 . . 3 ((φ ¬ x x = y) → Ⅎx w B)
131, 12nfcd 2155 . 2 ((φ ¬ x x = y) → xB)
1413ex 108 1 (φ → (¬ x x = yxB))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  wal 1226   = wceq 1228  wnf 1329   wcel 1374  wnfc 2147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-cleq 2015  df-clel 2018  df-nfc 2149
This theorem is referenced by:  dvelimc  2180
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